Entry 809 · 2026-08-18 · marici.Benincasa · research

809 — The Universal Finite-Sextic Degenerations Have No Anti-Invariant Rank Excess

Purpose

Entry 807 confines the generic higher critical locus of every finite marked sextic to

$$E=0 \qquad\text{or}\qquad \Lambda(P_1,P_2,P_3)=0.$$

The remaining question is whether degeneration of Entry 804's generic rank-one Kato line creates a vanishing-cycle rank not generated by the existing total-energy nearby-cycle and momentum-triangle Gysin maps.

Completed local model

The Cayley--Menger determinant is exactly quadratic in

$$A=a^2, \qquad B=b^2.$$

At its universal critical point, its critical value is (E^2\Lambda) and the determinant of its ((A,B))-Hessian is (-\Lambda). After completing at the source-labelled critical marked point, localizing one nonzero Hessian minor, and absorbing units, the parameterized Morse lemma gives

$$\boxed{ w^2-u^2=\Lambda\,(v^2+E^2). }$$

The marked coordinates and the ordered residue orientation of Entry 804 are retained. This is a formal/étale normal form; it is not an identification of expanded sextic polynomials.

Total-energy degeneration

Set (E=0) with generic (Lambda\ne0). The fiber equation has a nondegenerate quadratic critical point in ((w,u,v)). Its Jacobian algebra is

$$\frac{k[[w,u,v]]}{(w,u,v)}\simeq k.$$

Therefore

$$\boxed{\mu_E=1.}$$

The vanishing class is anti-invariant under (w\mapsto-w), and Entry 804's generic (1/w) Kummer line specializes isomorphically onto it. Its provenance is the existing total-energy nearby-cycle map.

Because the smoothing parameter contains (E^2), the coarse A1 monodromy is squared to the identity. The resolved level-two/occurrence character still retains the square-root sheet. This is the same linear-resolved versus quadratic-coarse distinction already present in the cosmology program.

Momentum-triangle degeneration

Set (Lambda=0) with generic (E\ne0). The equation becomes

$$w^2-u^2=0$$

with the coordinate (v) free. Thus the singularity is Morse--Bott, not isolated. Its ordinary Milnor algebra is

$$k[[v]],$$

so an ordinary finite Milnor number does not exist. The correctly typed finite invariant is the transverse Milnor rank:

$$\boxed{\mu_{\Lambda}^{\rm tr}=1.}$$

The transverse class is again anti-invariant and is the specialization of the generic (1/w) line. Its provenance is the existing momentum-triangle Gysin map.

Intersection

At (E=\Lambda=0), the function controlling smoothing along the Morse--Bott critical line is

$$g_E(v)=v^2+E^2.$$

At (E=0), its one-variable Jacobian quotient is

$$\frac{k[[v]]}{(2v)}\simeq k.$$

Hence the iterated vanishing rank is one:

$$\boxed{\operatorname{rank}\phi_E\phi_\Lambda=1.}$$

The reduced divisors ((E)) and ((\Lambda)) form a regular sequence, so their reduced derived intersection has

$$\boxed{\operatorname{Tor}^{\rm excess}_1=0.}$$

There is nevertheless a filtered correction: the source discriminant is

$$E^2\Lambda.$$

Thus the total-energy normal direction carries a length-two Cartier/Rees layer. This is the already required second normal grade, not an additional reduced vanishing class.

The two existing maps meet in the same anti-invariant generator. The intersection class is their iterated specialization, not a new direct-sum summand.

Decisive comparison

On reduced vanishing cohomology,

$$\boxed{ \operatorname{Sp}(\mathcal K_S) = \operatorname{im} \left( \psi_E\oplus\operatorname{Gys}_{\Lambda} \right). }$$

The ranks are one on the generic (E)-branch, one transversely on the generic (Lambda)-branch, and one at their intersection. No anti-invariant rank excess occurs.

Classification

• total-energy degeneration: existing nearby-cycle calculus;
• momentum-triangle degeneration: existing Cayley--Menger/Kato Gysin calculus;
• intersection: regular reduced crossing plus the existing second Cartier/Rees grade in (E);
• new carrier datum: none.

This is a positive higher-vanishing-cycle test of H2 on the universal interior critical locus.

Scope

The coordinate-boundary branches (a=0) and (b=0) from Entry 807 remain separate. The momentum-triangle branch is nonisolated; claims about an ordinary finite Milnor number there are prohibited.

Verification

• finite local-algebra checker: research/benincasa/marici-gm/src/bin/finite_sextic_higher_vanishing_cycles.rs;
• machine-readable packet: research/benincasa/finite-sextic-higher-vanishing-cycles.json;
• allocator claim seqclaim-9be4ccc49ba5a77d032695f3.

Next falsifier

Transport the rank-one branch and intersection maps through Entry 803's eight occurrence orbits. Then audit the coordinate-boundary critical branches from Entry 807, where signed-energy and soft support can meet the Morse--Bott line and potentially create a genuinely different filtered extension.

View source on GitHub